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If you have qualified for AIME, you have probably studied the more elementary AopS books fully, and can expect to get 4-6 or less with just these basics. If you want to score 7+ on the AIME, you will need to study the more advanced AoPS books listed above, as well as past AIME problems and solutions.

The official solution sets generally provide a single solution to a given problem, demonstrating the feasibility of solution within the standard high school curriculum. However, AIME questions are often quite rich and have many possible alternate solutions, which also should also be studied for effective contest preparation. The largest collection of alternate solutions are available on the AoPS web sites' AIME Problems and Solutions page at https://artofproblemsolving.com/wiki/index.php/AIME_Problems_and_Solutions

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This is the hard version of the problem. The only difference between the simple and hard versions is that in this version $$$u$$$ can take any possible value.

As is known, Omsk is the capital of Berland. Like any capital, Omsk has a well-developed metro system. The Omsk metro consists of a certain number of stations connected by tunnels, and between any two stations there is exactly one path that passes through each of the tunnels no more than once. In other words, the metro is a tree.

To develop the metro and attract residents, the following system is used in Omsk. Each station has its own weight $$$x \in \{-1, 1\}$$$. If the station has a weight of $$$-1$$$, then when the station is visited by an Omsk resident, a fee of $$$1$$$ burle is charged. If the weight of the station is $$$1$$$, then the Omsk resident is rewarded with $$$1$$$ burle.

Omsk Metro currently has only one station with number $$$1$$$ and weight $$$x = 1$$$. Every day, one of the following events occurs:

• A new station with weight $$$x$$$ is added to the station with number $$$v_i$$$, and it is assigned a number that is one greater than the number of existing stations.
• Alex, who lives in Omsk, wonders: is there a subsegment$$$\dagger$$$ (possibly empty) of the path between vertices $$$u$$$ and $$$v$$$ such that, by traveling along it, exactly $$$k$$$ burles can be earned (if $$$k < 0$$$, this means that $$$k$$$ burles will have to be spent on travel). In other words, Alex is interested in whether there is such a subsegment of the path that the sum of the weights of the vertices in it is equal to $$$k$$$. Note that the subsegment can be empty, and then the sum is equal to $$$0$$$.

You are a friend of Alex, so your task is to answer Alex's questions.

$$$\dagger$$$Subsegment — continuous sequence of elements.

The first line contains a single number $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

The first line of each test case contains the number $$$n$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$) — the number of events.

Then there are $$$n$$$ lines describing the events. In the $$$i$$$-th line, one of the following options is possible:

• First comes the symbol "+" (without quotes), then two numbers $$$v_i$$$ and $$$x_i$$$ ($$$x_i \in \{-1, 1\}$$$, it is also guaranteed that the vertex with number $$$v_i$$$ exists). In this case, a new station with weight $$$x_i$$$ is added to the station with number $$$v_i$$$.
• First comes the symbol "?" (without quotes), and then three numbers $$$u_i$$$, $$$v_i$$$, and $$$k_i$$$ ($$$-n \le k_i \le n$$$). It is guaranteed that the vertices with numbers $$$u_i$$$ and $$$v_i$$$ exist. In this case, it is necessary to determine whether there is a subsegment (possibly empty) of the path between stations $$$u_i$$$ and $$$v_i$$$ with a sum of weights exactly equal to $$$k_i$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

For each of Alex's questions, output "Yes" (without quotes) if the subsegment described in the condition exists, otherwise output "No" (without quotes).

You can output the answer in any case (for example, the strings " yEs ", " yes ", " Yes " and " YES " will be recognized as a positive answer).

Explanation of the first sample.

The answer to the second question is "Yes" , because there is a path $$$1$$$.

In the fourth question, we can choose the $$$1$$$ path again.

In the fifth query, the answer is "Yes" , since there is a path $$$1-3$$$.

In the sixth query, we can choose an empty path because the sum of the weights on it is $$$0$$$.

It is not difficult to show that there are no paths satisfying the first and third queries.

Approximate solution of the control problem of supplies with many intervals and concave cost functions

• Control in Social Economic Systems
• Published: 18 July 2008
• Volume 69 , pages 1181–1187, ( 2008 )

Cite this article

• A. V. Eremeev 1 ,
• M. Ya. Kovalyov 2 &
• P. M. Kuznetsov 3  

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The problem of searching for a cost-minimum plan of supplies of uniform products to one consumer is considered. The set of admissible intervals of the supply volume and concave cost functions of supplies within each interval are preassigned for each supplier. The totally polynomial ɛ -approximate algorithm for the given problem and the pseudopolynomial exact algorithm for its partial case are suggested.

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Eremeev, A.V., Romanova, A.A., Servakh, V.V., and Chauhan, S.S., Approximate Solution of One Control Problem of Supplies, Diskret. Analiz Issl. Operats. , 2006, series 2, vol. 13, no. 1, pp. 27–39.

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Omsk Affiliated Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Omsk, Russia

A. V. Eremeev

Belarussian State University, Minsk, Belarus

M. Ya. Kovalyov

Omsk State University, Omsk, Russia

P. M. Kuznetsov

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Original Russian Text © A.V. Eremeev, M.Ya. Kovalyov, P.M. Kuznetsov, 2008, published in Avtomatika i Telemekhanika, 2008, No. 7, pp. 90–97.

This work was supported by the program INTAS, project no. 03-51-5501. Investigations of M.Ya. Kovalyov were performed in the framework of the State program of scientific investigations of Belorussia “Mathematical models.”

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Eremeev, A.V., Kovalyov, M.Y. & Kuznetsov, P.M. Approximate solution of the control problem of supplies with many intervals and concave cost functions. Autom Remote Control 69 , 1181–1187 (2008). https://doi.org/10.1134/S0005117908070096

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Received : 22 May 2007

Published : 18 July 2008

Issue Date : July 2008

DOI : https://doi.org/10.1134/S0005117908070096

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